Solving Some Definite Integrals by Using Maple

Transcription

1 Wold Joul of Coute Alictio d Techology 3: 6-65, DOI:.389wct..3 htt: Solvig Soe Defiite Itegls y Usig Mle Chii-Huei Yu Detet of Mgeet d Ifotio, N Jeo Uivesity of Sciece d Techology, Ti City, 7376, Tiw *Coesodig Autho: chiihuei@il.tc.edu.tw Coyight Hoizo Resech Pulishig All ights eseved. Astct This study uses the theticl softwe Mle fo the uiliy tool to evlute two tyes of defiite itegls. We c oti the closed fos of these two tyes of defiite itegls y usig diffeetitio with esect to ete d Leiiz diffeetil ule. At the se tie, we ovide soe defiite itegls to do clcultio cticlly. The esech ethods doted i this study ivolved fidig solutios though ul clcultios d veifyig these solutios y usig Mle. Keywods Defiite Itegls, Closed Fos, Diffeetitio With Resect to Pete, Leiiz Diffeetil Rule, Mle. Itoductio As ifotio techology dvces, whethe coutes c ecoe cole with hu is to efo stct tss, such s stct t siil to the itigs of Picsso d usicl coositios siil to those of Beethove, is tul uestio. Cuetly, this es uttile. I dditio, whethe coutes c solve stct d difficult theticl oles d develo stct theticl theoies such s those of theticis lso es ufesile. Nevetheless, i seeig fo ltetives, we c study wht ssistce theticl softwe c ovide. This study itoduces how to coduct theticl esech usig the theticl softwe Mle. The i esos of usig Mle i this study e its sile istuctios d ese of use, which ele egies to le the oetig techiues i shot eiod. By eloyig the oweful coutig cilities of Mle, difficult oles c e esily solved. Eve whe Mle cot deteie the solutio, ole-solvig hits c e idetified d ifeed fo the oite vlues clculted d solutios to siil oles, s deteied y Mle. Fo this eso, Mle c ovide isights ito scietific esech. Iuiig though olie suot syste ovided y Mle o owsig the Mle wesite c fcilitte futhe udestdig of Mle d ight ovide ueected isights. Fo the istuctios d oetios of Mle, [-7 c e doted s efeeces. I clculus d egieeig thetics couses, we let y ethods to solve the itegl oles, icludig chge of viles ethod, itegtio y ts ethod, til fctios ethod, tigooetic sustitutio ethod, d so o. I this e, we study the evlutio of the followig two tyes of defiite itegls which e ot esy to oti thei swes usig the ethods etioed ove. d si d si whee,,, e o-egtive iteges,,,,, e el ues, d >, >. We c oti the closed fos of these two tyes of defiite itegls y usig diffeetitio with esect to ete d Leiiz diffeetil ule ; these e the i esults of this e i.e., Theoes,. The study of elted itegl oles c efe to [8-6. Siulteously, we c coe the ethod used i this ticle with tht i [7. O the othe hd, we ovide soe defiite itegls to do clcultio cticlly. The esech ethods doted i this study ivolved fidig solutios though ul clcultios d veifyig these solutios y usig Mle. This tye of esech ethod ot oly llows the discovey of clcultio eos, ut lso hels odify the oigil diectios of thiig fo ul d Mle clcultios. Fo this eso, Mle ovides isights d guidce egdig ole-solvig ethods.. Mi Results Fistly, we itoduce soe ottios d foul used i this study... Nottios i the -th ode deivtive of the fuctio u

2 6 Solvig Soe Defiite Itegls y Usig Mle is deoted y u,whee is o-egtive itege. ii Suose s, t e el ues, we defie s t s s s t, d s. iii Suose, e ositive iteges, we defie.. Itegl Foul [8 Suose, d., e el ues d >,the the defiite itegl d. Net, we itoduce two iott theoes used i this e..3. Diffeetitio With Resect to Pete [9 Suose c,d,, e el ues d the fuctio f, is defied o [ c, d [,. If f, d its f til deivtive, e cotiuous fuctios o [ c, d [,. The F f, d is diffeetile d o the oe itevl c, d, d its deivtive F d f, d fo ll c, d... Leiiz Diffeetil Rule [3 Suose is o-egtive itege, f d g e -ties diffeetile fuctios. The the -th ode deivtive of the oduct fuctio f g, f g f g The followig is the fist esult of this study, we oti the closed fo of defiite itegl..5. Theoe Assue, e o-egtive iteges,, e el ues, d >. The the defiite itegl d.5.. Poof 3 By the itegl foul, we oti d Usig diffeetitio with esect to ete d Leiiz diffeetil ule, diffeetitig -ties with esect to o oth sides of, we hve d [ [ Theefoe, d 6 Also, y diffeetitio with esect to ete d Leiiz diffeetil ule, diffeetitig -ties with esect to o oth sides of 6, we oti d [ [ Thus, 5 7

3 Wold Joul of Coute Alictio d Techology 3: 6-65, 63 d Net, we deteie the closed fo of defiite itegl..6. Theoe If,, e o-egtive iteges,,, e el ues, d >. The the defiite itegl d si si Poof Becuse d si d d [ d [ y 6 9 Also, usig diffeetitio with esect to ete d Leiiz diffeetil ule, diffeetitig -ties with esect to o oth sides of 9, we oti d si Agi, y diffeetitio with esect to ete d Leiiz diffeetil ule, diffeetitig -ties with esect to o oth sides of, the d si si 3. Eles I the followig, fo the two tyes of defiite itegls i this study, we oose soe eles d use Theoes d to deteie thei closed fos. O the othe hd, we eloy Mle to clculte the oitios of these defiite itegls d thei closed fos fo veifyig ou swes. 3.. Ele I Theoe, tig 3 5,,,, we oti the followig defiite itegl d Usig Mle to veify the coectess of. >evlfit^53*^7,..pi,; >evlfpi5*susuoduct--,..-*oduct- -s,s..-*oduct---t,t..-*oduct-5 -w,w..3-*-**-*-^*8^---* ^-3,..,..,;

4 6 Solvig Soe Defiite Itegls y Usig Mle 3.. Ele I Theoe, let 5, 3, 7,, the the defiite itegl 5 d Net, we use Mle to veify the coectess of. >evlfit^57-*^9,..pi,; >evlf-pi56*susuoduct--,..-*oduct --s,s..-*oduct---t,t..-*oduct -7-w,w..-*3-**5-*-^*5^--*9^-7,..3,..5,; 3.3. Ele I Theoe, let,,,, 3,, we c deteie the defiite itegl si d 8 3 si >evlfit^*si^3*^*si ^^8,..Pi,; >evlf-6*pi5*suoduct--,..-*oduct --s,s..3-*oduct---t,t..*-9*- *^-5-*8^-,..,; 3.. Ele I Theoe, if 3,,, 5,,, the the defiite itegl 8 si d 5 si >evlfit^*si^85-*^*si ^^,..Pi,; >evlf-8*pi95*suoduct--,..-*oduct- -s,s..-*oduct---t,t..*oduct-7-w,w..3*3-*6^-5-*8^-5,..3,;. Coclusio I this e, we ovide ew techiue to solve two tyes of defiite itegls, d we ow tht diffeetitio with esect to ete d Leiiz diffeetil ule ly sigifict oles i the theoeticl ifeeces of this study. I fct, the lictios of these two theoes e etesive, d c e used to esily solve y difficult oles; we edevo to coduct futhe studies o elted lictios. O the othe hd, Mle lso lys vitl ssistive ole i ole-solvig. I the futue, we will eted the esech toic to othe clculus d egieeig thetics oles d solve these oles y usig Mle. These esults will e used s techig teils fo Mle o eductio d esech to ehce the coottios of clculus d egieeig thetics. REFERENCES [ F. Gv, The Mle Boo, Lodo: Ch & HllCRC,. [ M. L. Aell d J. P. Bselto, Mle y Ele, 3d ed., New Yo: Elsevie Acdeic Pess, 5. [3 D. Richds, Advced Mtheticl Methods with Mle, New Yo: Cidge Uivesity Pess,. [ C. Tocci d S. G. Ads, Alied Mle fo Egiees d Scietists, Bosto: Atech House, 996. [5 J. S. Roetso, Egieeig Mthetics with Mle, New Yo: McGw-Hill, 996. [6 R. J. Stoee d J. F. Kshoe, Discoveig Mthetics with Mle : A Itective Elotio fo Mtheticis, Egiees d Ecooeticis, Bsel: Bihuse Velg, 999. [7 C. T. J. Dodso d E. A. Gozlez, Eeiets i Mthetics Usig Mle, New Yo: Sige-Velg, 995. [8 A. A. Ads, H. Gottliese, S. A. Lito, d U. Mti, Autoted theoe ovig i suot of coute lge: syolic defiite itegtio s cse study, Poceedigs of the 999 Itetiol Syosiu o Syolic d Algeic Couttio,. 53-6, Vcouve, Cd, 999. [9 C. Oste, Liit of defiite itegl, SIAM Review, Vol. 33, No.,. 5-6, 99. [ M. A. Nylo, O the evlutio of defiite itegl

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